What is topological matter & why do we care? Part 1: what are topological insulators?

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What is topological matter

&

why do we care?

Part 1: what are topological insulators?

Eddy Ardonne thanks to:

Hans Hansson

WfSW:QT 2014-08-29

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Topological insulators in the media!

Fourth season of Big Bang theory: ‘The thespian catalyst’.

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Topological insulators in the media!

Fourth season of Big Bang theory: ‘The thespian catalyst’.

So, who knows what a topological insulator is?

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Condensed matter physics

What phases of matter do exist?

How does matter go from one phase to another?

Daily life example: water and ice

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Condensed matter physics

Daily life example: water and ice Other examples: magnets

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Condensed matter physics

Daily life example: water and ice Other examples: superconductors

Kamerling Onnes et al., 1911

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Condensed matter physics

Some materials (such as copper) are metallic, they conduct current very well:

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Condensed matter physics

Some materials (such as wood) are insulating, they do not conduct current:

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Condensed matter physics

Some materials (such as wood) are insulating, they do not conduct current:

Both conductors and insulators are important, but the

behaviour in between is really interesting: semi-conductors!

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Between metals and insulators

Semiconductors can be used to make very interesting devices, transistors!

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Between metals and insulators

Bardeen et al., 1947

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Between metals and insulators

Impact on society is hard to quantify.

Bardeen et al., 1947

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Topological insulators in words

A topological insulator is also in between metals and insulators.

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Topological insulators in words

They are insulating inside (in ‘the bulk’)

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Topological insulators in words

They are conducting on the surface

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Topological insulators in words

They are conducting on the surface

The conductance on the surface is insensitive to dirt, disturbances, etc.

We say that the conductance is protected for topological reasons!

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Crash course on topology

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Crash course on topology

A football player would say:

≠

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Crash course on topology

≠

A topologist would say:

=

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Crash course on topology

≠

=

≠

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Knot theory, some history

Inspired by Peter Tait’s experiments on smoke rings, Sir William Thomson developed the idea that the different atoms are related to different knots

(1867)!

The knots were thought to be different vortex rings in the aether. Tait started to classify all the different knots. However, the aether doesn’t exist, so this idea failed.

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Knot theory, some history

(1867)!

Nevertheless, telling knots apart is an interesting and difficult problem.

No topological invariant that can distinguish all knots is known today.

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Knot theory, some history

(1867)!

Nevertheless, telling knots apart is an interesting and difficult problem.

No topological invariant that can distinguish all knots is known today.

Knots play an important role in topological quantum computation!

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Knots, some examples

For ‘small’ knots, it’s rather easy to tell them apart

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Knots, some examples

But are these the same or not?

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Knots, some examples

But are these the same or not?

Yes, but for a long time they were listed as different in the literature!

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Topological invariant: winding number

An example of a topological invariant is the winding number.

How many times winds a curve around the origin?

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Topological invariant: winding number

-2 -1 1 2 3

-2 -1 1 2

start

finish winding number: +2

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Topological invariant: winding number

start

finish winding number: not defined

corresponds to a phase transition

-1.0 -0.5 0.5 1.0 1.5 2.0

-1.5 -1.0 -0.5 0.5 1.0 1.5

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Topological invariant: winding number

start

finish winding number: -1

-0.5 0.5 1.0 1.5

-1.0 -0.5 0.5 1.0

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The quantum Hall effect

The first topological state that was observed is the ‘quantum Hall effect’, dating back to 1980.

The electrons are confined to a two-dimensional plane, between to semiconductors. The quantum Hall effect occurs at very low temperatures, 1 Kelvin or lower, and in a strong magnetic field, 10 Tesla.

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The quantum Hall effect

The first topological state that was observed is the ‘quantum Hall effect’, dating back to 1980.

The electrons are confined to a two-dimensional plane, between to semiconductors. The quantum Hall effect occurs at very low temperatures, 1 Kelvin or lower, and in a strong magnetic field, 10 Tesla.

Charged particles that move in a magnetic field experience a ‘Lorentz force’, perpendicular to the field and direction they move in.

Example: earth magnetic field, giving rise to the northern light:

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The classical Hall effect

B

−

V_H

V_L

x y

z

When there is a current through a I

thin strip, there is voltage drop along the current.

In a magnetic field, there is also a voltage perpendicular to the

current, the Hall voltage.

The resistance is the ratio of the

voltage to the current: ^R^L ⁼

V_L

I R_H = V_H

I The (classical) Hall resistance is

proportional to the magnetic field:

Magnetic field

Hall resistance

Classical Hall effect

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The quantum Hall effect

^B

−

V_H

V_L

x y

z

I

In very clean samples, at low temperatures, and high fields, the Hall conductance becomes

quantized:

H = 1/R_H = ⌫ e² h

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The quantum Hall effect

^B

−

V_H

V_L

x y

z

I

quantized:

H = 1/R_H = ⌫ e² h

Here, e is the charge of the electron, h is Planck’s constant, and ν is an integer. This quantization is better than one in a billion (and agrees with particle accelerator experiments)!

Red curve: Hall resistance

Green curve: longitudinal resistance

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The quantum Hall effect

^B

−

V_H

V_L

x y

z

I

quantized:

H = 1/R_H = ⌫ e² h

Here, e is the charge of the electron, h is Planck’s constant, and ν is an integer. This quantization is better than one in a billion (and agrees with particle accelerator experiments)!

Red curve: Hall resistance

Green curve: longitudinal resistance Hall resistance is a topological

invariant

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The quantum Hall effect

Inside the two dimensional droplet, the electrons are localized, and move in small circles (Larmor orbitals). However, electrons can move along the

boundary, but only in one direction.

quantum Hall insulator vacuum

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The quantum Hall effect

Inside the two dimensional droplet, the electrons are localized, and move in small circles (Larmor orbitals). However, electrons can move along the

boundary, but only in one direction.

quantum Hall insulator vacuum

The quantum Hall liquid is an insulator, with a chiral edge mode!

The electrons on the edge can only move in one direction, which can not be changed, not even by dirt, disorder, etc. They can simply not turn back. This is why the quantization is so precise!

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Are there other topological states?

The quantum Hall effect is very interesting, and it’s used as a standard for resistance.

But it requires a very high magnetic field, which makes it unsuitable for technological applications.

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Are there other topological states?

It was long believed that the magnetic field is necessary. Without it, there are two edge modes, moving in opposite directions. These modes can interact with each other, destroying the topological properties and the quantized conductance.

Kane and Mele (UPenn) realized in 2005 that there is a way around this problem.

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Are there other topological states?

It was long believed that the magnetic field is necessary. Without it, there are two edge modes, moving in opposite directions. These modes can interact with each other, destroying the topological properties and the quantized conductance.

Kane and Mele (UPenn) realized in 2005 that there is a way around this problem.

To explain this, we have to make a step back, and look at the difference between metals and insulators.

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Energy levels

The energy levels of atoms, obtained by solving the Schrödinger equation, are discrete. For instance, for the hydrogen atom.

H = E

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Band theory

What happens if the electrons move in a periodic potential, such as in solids?

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Band theory

Bloch theorem says that the energy levels form continuous bands of

states, that are periodic:

(~r) = e ^i~ ^k ^·~r u(~r)

periodic function

‘plane’ wave

The energy of the states depends smoothly on the ‘wave vector’ k, forming a ‘band’, as opposed to discrete levels.

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Band theory

Bloch theorem says that the energy levels form continuous bands of

states, that are periodic:

(~r) = e ^i~ ^k ^·~r u(~r)

periodic function

‘plane’ wave

The energy of the states depends smoothly on the ‘wave vector’ k, forming a ‘band’, as opposed to discrete levels.

To what level the bands are filled, determines the properties of solids!

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Metals v.s. insulators

momentum

energi

valence band conduction band

(empty) band insulator

Fermi level

momentum

energi

metal

Fermi level

Often, there is a gap between different bands. To which energy one has to fill each band (called the Fermi level), depends on the number of

electrons that are not bound to an atom.

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Metals v.s. insulators

momentum

energi

Fermi level

momentum

energi

metal

Fermi level

In a metal, applying an electric fields give electrons slightly more energy, allowing it to conduct.

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Metals v.s. insulators

momentum

energi

Fermi level

momentum

energi

metal

Fermi level

In a metal, applying an electric fields give electrons slightly more energy, allowing it to conduct.

In an insulator, the only way to give an electron more energy, is to

excite it over the gap, which costs a lot of energy. Therefore, insulators do not conduct for low electric fields!

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Quantum Hall effect revisited

Using band theory, we can now also explain the situation in the quantum Hall effect.

There is a single energy level that connects the valence and conduction band. As long as the Fermi level is in the gap, there is a state that

conducts!

The slope of the level gives the direction in which the electron moves, so we also find that this level is chiral.

momentum

energi

chiral edge mode quantum Hall insulator

Fermi level

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Kane & Mele’s model

Kane and Mele realized that topological phases can exist, even without a strong magnetic field.

In there model, there are two edge modes, moving in opposite directions.

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Kane & Mele’s model

Electrons have spin, which can be up, down, or in an arbitrary superposition.

|electron >= ↵| "> + | #>

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Kane & Mele’s model

In the absence of magnetic fields, energy levels come in pairs, with exactly the same energy.

Magnetic fields can be external, or due to the motion of the electrons themselves, called spin-orbit coupling.

Electrons have spin, which can be up, down, or in an arbitrary superposition.

|electron >= ↵| "> + | #>

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Kane & Mele’s model

For certain values of the momentum, there are no magnetic fields, so the energy levels must come in pairs! Example: point A.

momentum

energi

A

topological insulator

Fermi level

momentum

energi

A

trivial insulator

Fermi level

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Kane & Mele’s model

For certain values of the momentum, there are no magnetic fields, so the energy levels must come in pairs! Example: point A.

momentum

energi

A

topological insulator

Fermi level

momentum

energi

A

trivial insulator

Fermi level

There are two, topologically distinct ways of drawing the diagram!

So there are two types of insulators, one with a pair of edge modes!

These modes are chiral, and move in opposite direction.

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Topological insulators: CdTe v.s. HgTe

Kane and Mele worked on graphene, which is always trivial.

Zhang & coworkers designed an experiment:

CdTe is trivial, while HgTe is topological

CdTe HgTe

Zhang et al., 2006 (Picture: Physics Today)

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Topological insulators: CdTe v.s. HgTe

Zhang & coworkers designed an experiment:

Take a sandwich of CdTe-HgTe-CdTe, and vary the thickness!

Zhang et al., 2006 (Picture: Physics Today)

Trivial case Topologial case

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Topological insulators: CdTe v.s. HgTe

Molenkamp et al., 2007 (Picture: Physics Today)

Trivial case Topologial case

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Topological insulators: CdTe v.s. HgTe

Trivial case Topologial case

The measurements show that the conductance is quantized in the topological case.

Other experiments show that the system indeed has chiral edge modes.

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Topological insulators: CdTe v.s. HgTe

Trivial case Topologial case

The measurements show that the conductance is quantized in the topological case.

Other experiments show that the system indeed has chiral edge modes.

Conclusion: this really is a topological insulator!

By now, many types are observed experimentally, in several groups!

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Conclusions part I

In condensed matter, often experiments come (way) before the theory (superconductivity, quantum Hall effect for instance).

Topological insulators were predicted on theoretical grounds, and observed afterwards.

The topology in the problem allows for very precise predictions, that can be verified.

Topological insulators are extremely interesting from a fundamental perspective.

Can topological insulators be used for devices?

What is topological matter & why do we care? Part 1: what are topological insulators?